
from __future__ import annotations

from scipy.special import comb  # type: ignore

class 							_UpperCamelCase					:






			'''simple docstring'''

			def __init__(							self			:							Optional[Any]    ,						snake_case_			:							list[tuple[float, float]]   ):
							UpperCamelCase_:				int						=    list_of_points
							# Degree determines the flexibility of the curve.
							# Degree = 1 will produce a straight line.
							UpperCamelCase_:				Union[str, Any]						=    len(snake_case_   ) - 1

			def    lowerCAmelCase__						(							self			:							Optional[int]    ,						snake_case_			:							float   ):
							assert 0 <= t <= 1, "Time t must be between 0 and 1."
							UpperCamelCase_:				list[float]						=    []
							for i in range(len(self.list_of_points   )   ):
											# basis function for each i
											output_values.append(
											    comb(self.degree    ,						snake_case_   ) * ((1 - t) ** (self.degree - i)) * (t**i)   )
							# the basis must sum up to 1 for it to produce a valid Bezier curve.
							assert round(sum(snake_case_   )    ,						5   ) == 1
							return output_values

			def    lowerCAmelCase__						(							self			:							Any    ,						snake_case_			:							float   ):
							assert 0 <= t <= 1, "Time t must be between 0 and 1."

							UpperCamelCase_:				str						=    self.basis_function(snake_case_   )
							UpperCamelCase_:				List[Any]						=    0.0
							UpperCamelCase_:				int						=    0.0
							for i in range(len(self.list_of_points   )   ):
											# For all points, sum up the product of i-th basis function and i-th point.
											x += basis_function[i] * self.list_of_points[i][0]
											y += basis_function[i] * self.list_of_points[i][1]
							return (x, y)




			def    lowerCAmelCase__						(							self			:							Optional[Any]    ,						snake_case_			:							float = 0.01   ):
							from matplotlib import pyplot as plt  # type: ignore

							UpperCamelCase_:				list[float]						=    []  # x coordinates of points to plot
							UpperCamelCase_:				list[float]						=    []  # y coordinates of points to plot

							UpperCamelCase_:				List[Any]						=    0.0
							while t <= 1:
											UpperCamelCase_:				Optional[Any]						=    self.bezier_curve_function(snake_case_   )
											to_plot_x.append(value[0]   )
											to_plot_y.append(value[1]   )
											t += step_size

							UpperCamelCase_:				Dict						=    [i[0] for i in self.list_of_points]
							UpperCamelCase_:				List[Any]						=    [i[1] for i in self.list_of_points]

							plt.plot(
							    snake_case_    ,						snake_case_    ,						color="""blue"""    ,						label="""Curve of Degree """ + str(self.degree   )    ,						)
							plt.scatter(snake_case_    ,						snake_case_    ,						color="""red"""    ,						label="""Control Points"""   )
							plt.legend()
							plt.show()


if __name__ == "__main__":
							import doctest

							doctest.testmod()

							BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
							BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
							BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3








from __future__ import annotations







def 							A__       (    lowerCamelCase    )	->						bool:
				UpperCamelCase_:				Optional[int]						=    len(lowerCamelCase    )
				# We need to create solution object to save path.
				UpperCamelCase_:				List[str]						=    [[0 for _ in range(lowerCamelCase    )] for _ in range(lowerCamelCase    )]
				UpperCamelCase_:				Dict						=    run_maze(lowerCamelCase  ,     0  ,     0  ,     lowerCamelCase    )
				if solved:
								print("""\n""".join(str(lowerCamelCase    ) for row in solutions    )    )
				else:
								print("""No solution exists!"""    )
				return solved







def 							A__       (    lowerCamelCase  ,     lowerCamelCase  ,     lowerCamelCase  ,     lowerCamelCase    )	->						bool:
				UpperCamelCase_:				Dict						=    len(lowerCamelCase    )
				# Final check point.
				if i == j == (size - 1):
								UpperCamelCase_:				Union[str, Any]						=    1
								return True

				UpperCamelCase_:				List[str]						=    (not i < 0) and (not j < 0)  # Check lower bounds
				UpperCamelCase_:				List[str]						=    (i < size) and (j < size)  # Check upper bounds

				if lower_flag and upper_flag:
								# check for already visited and block points.
								UpperCamelCase_:				Dict						=    (not solutions[i][j]) and (not maze[i][j])
								if block_flag:
												# check visited
												UpperCamelCase_:				Optional[Any]						=    1

												# check for directions
												if (
												    run_maze(lowerCamelCase  ,     i + 1  ,     lowerCamelCase  ,     lowerCamelCase    )
												    or run_maze(lowerCamelCase  ,     lowerCamelCase  ,     j + 1  ,     lowerCamelCase    )
												    or run_maze(lowerCamelCase  ,     i - 1  ,     lowerCamelCase  ,     lowerCamelCase    )
												    or run_maze(lowerCamelCase  ,     lowerCamelCase  ,     j - 1  ,     lowerCamelCase    )
												):
																return True

												UpperCamelCase_:				Optional[Any]						=    0
												return False
				return False


if __name__ == "__main__":
							import doctest

							doctest.testmod()






